New Stability Conditions for Linear Difference Equations using Bohl-Perron Type Theorems
aa r X i v : . [ m a t h . D S ] J un New Stability Conditions for Linear DifferenceEquations using Bohl-Perron Type Theorems
Leonid Berezansky Department of Mathematics, Ben-Gurion University of the Negev,Beer-Sheva 84105, Israeland Elena Braverman Department of Mathematics and Statistics, University of Calgary,2500 University Drive N.W., Calgary, AB T2N 1N4, Canada
AMS Subject Classification:
Key words:
Linear delay difference equations, exponential stability, positive fundamen-tal function.
Abstract
The Bohl-Perron result on exponential dichotomy for a linear difference equation x ( n + 1) − x ( n ) + m X l =1 a l ( n ) x ( h l ( n )) = 0 , h l ( n ) ≤ n, states (under some natural conditions) that if all solutions of the non-homogeneousequation with a bounded right hand side are bounded, then the relevant homogeneousequation is exponentially stable. According to its corollary, if a given equation is close to an exponentially stable comparison equation (the norm of some operator is less thanone), then the considered equation is exponentially stable.For a difference equation with several variable delays and coefficients we obtainnew exponential stability tests using the above results, representation of solutions andcomparison equations with a positive fundamental function. In this paper we study stability of a scalar linear difference equation with several delays x ( n + 1) − x ( n ) = − m X l =1 a l ( n ) x ( h l ( n )) , h l ( n ) ≤ n, (1) Partially supported by Israeli Ministry of Absorption Partially supported by the NSERC Research Grant h l ( n ) is an integer for any l = 1 , · · · , m and n = 0 , , , · · · under the following tworestrictions on the parameters of (1) which mean that coefficients and delays are bounded: (a1) there exists K > | a l ( n ) | ≤ K for l = 1 , · · · , m , n = 0 , , , · · · ; (a2) there exists T > n − T ≤ h l ( n ) ≤ n for l = 1 , · · · , m , n = 0 , , , · · · .Stability of equation (1) has been an intensively developed area during the last twodecades, see [2, 4], [8]-[10], [12, 13], [15]-[24] and references therein.In the present paper we study a connection between stability and existence of a positivesolution for a general linear difference equation with nonnegative variable coefficients andseveral delays. This idea was developed in [12] for equations with constant coefficients, seealso [20] and references therein for some further results on equations with variable coefficientsand a nonlinear part.The method of the present paper is based on the Bohl-Perron type theorems whichconnect the boundedness of all solutions for all bounded right hand sides with the exponentialstability of the relevant homogeneous equation. Here we apply the development of thismethod [1, 2] where stability properties of the original equation are established based onthe known asymptotics of an auxiliary (comparison, model) equation (see Lemma 1 below).This idea can be compared to [20] where a nonlinear perturbation of a linear equation isconsidered. The difference between the present paper and [2] is that the model equation is,generally, a high order equation, which allows to consider equations with large delays.The paper is organized as follows. In Section 2 we deduce some general exponentialstability results for high order difference equations with variable coefficients. Section 3presents explicit stability tests in terms of delays and coefficients. Finally, Section 4 involvesdiscussion and examples, which compare our results with known stability tests, and outlinesopen problems. We assume that (a1)-(a2) hold for equation (1) and all other equations in the paper. Inparticular, the system has a finite prehistory: h l ( n ) ≥ n − T , n ≥ n , for any n ≥
0. We note that, generally, under this assumption (1) can be written as a higher orderequation x ( n + 1) − x ( n ) = − T X l =0 b l ( n ) x ( n − l ), where b l ( n ) are defined by coefficients a k ( n )of equation (1). However, there are several reasons why we prefer form (1).1. The number of nonzero terms in (1) can be much smaller than T ; this form allowsus to specify stability conditions for equations with one, two or three variable delays,equations with positive and negative coefficients (where positive and negative termsare written separately) and so on. In particular, we refer to the long tradition when,for example, the equation with one variable delay x ( n + 1) − x ( n ) = a ( n ) x ( h ( n )) wasstudied in this form.2. The main idea of the method applied in the present paper is based on the comparison2o a model stable equation. Form (1) gives some flexibility: in many cases, the modelequation may have different delays compared to the original equation.In future, we will consider the scalar linear difference equation x ( n + 1) − x ( n ) = − m X l =1 a l ( n ) x ( h l ( n )) + f ( n ) , n ≥ n , (2) x ( n ) = ϕ ( n ) , n ≤ n , (3)and the corresponding homogeneous equation (1).Further we will extensively apply the solution representation formula and properties ofthe fundamental function. We start with the definition of this function. Definition . The solution X ( n, k ) of the problem x ( n + 1) − x ( n ) = − m X l =1 a l ( n ) x ( h l ( n )) , n > k, x ( n ) = 0 , n < k, x ( k ) = 1is called the fundamental function of equation (2) (or of (1) ).In the following we will use a modification of the solution representation formula [7] x ( n ) = X ( n, n ) x ( n ) + n − X k = n X ( n, k + 1) f ( k ) − n − X k = n X ( n, k + 1) m X l =1 a l ( k ) ϕ ( h l ( k )) , (4)where ϕ ( h l ( k )) = 0 , h l ( k ) ≥ n .Denote by l ∞ the space of bounded sequences v = { x ( n ) } with the norm k v k l ∞ = sup n ≥ | x ( n ) | < ∞ . Lemma 1 [ ] Suppose for the equation L ( { x ( n ) } ) = 0 , where L ( { x ( n ) } ∞ n =0 ) = ( x ( n + 1) − x ( n ) + m X l =1 a l ( n ) x ( h l ( n )) ) ∞ n =0 , conditions (a1)-(a2) hold, the equation L ( { x ( n ) } ) = 0 is exponentially stable, where L ( { x ( n ) } ∞ n =0 ) = ( x ( n + 1) − x ( n ) + r X l =1 b l ( n ) x ( g l ( n )) ) ∞ n =0 . (5) Let Y ( n, k ) be the fundamental function of (5) and C be the Cauchy operator of equation(5) which is C { f ( n ) } ∞ n =0 = ( y ( n ) = n − X k =0 Y ( n, k + 1) f ( k ) ) ∞ n =0 , (6) with y (0) = 0 . If the operator T = C ( L − L ) = I − C L satisfies kT k l ∞ → l ∞ < , then theequation L ( { x ( n ) } ) = 0 is exponentially stable. efinition. Equation (1) is exponentially stable if there exist constants
M > λ ∈ (0 , { x ( n ) } of (1),(3) the inequality | x ( n ) | ≤ M λ n − n (cid:18) max − T + n ≤ k ≤ n {| ϕ ( k ) |} (cid:19) (7)holds for all n ≥ n , where M, λ do not depend on n .The following result states that the exponential estimate of the fundamental functionimplies exponential stability (see [2], Theorem 4). Lemma 2 [ ] Suppose there exist
L, µ , L > , < µ < , such that | X ( n, k ) | ≤ Lµ n − k , n ≥ k ≥ n . (8) Then there exist
M, λ , M > , < λ < such that inequality (7) holds for a solution of (1),(3). Conversely, if (7) holds, then (8) is also valid, with L = M , µ = λ . We proceed to estimation of the fundamental function. Let us note that throughout thepaper we assume that the sum equals zero and the product equals one if the lower boundexceeds the upper bound.
Lemma 3
The fundamental function of equation (1) has the following estimate | X ( n, k ) | ≤ n − Y j = k m X l =1 | a l ( j ) | ! . Proof.
Let us fix k and denote x ( n ) = X ( n, k ). By definition we have x ( n ) = x ( n − − m X l =1 a l ( n − x ( h l ( n − ,x ( n ) = 0 , n < k, x ( k ) = 1 . Then | x ( n ) | ≤ | x ( n − | + m X l =1 | a l ( n − | | x ( h l ( n − | . Consider the linear equation y ( n ) = y ( n −
1) + m X l =1 | a l ( n − | y ( h l ( n − y ( n ) = 0 , n < k, y ( k ) = 1 .
4y induction it is easy to see that { y ( n ) } is a positive nondecreasing sequence and | x ( n ) | ≤ y ( n ), so y ( h l ( n − ≤ y ( n − y ( n ) ≤ y ( n − m X l =1 | a l ( n − | ! ≤ n − Y j = k m X l =1 | a l ( j ) | ! y ( k ) = n − Y j = k m X l =1 | a l ( j ) | ! . The remark that | X ( n, k ) | = | x ( n ) | ≤ y ( n ) completes the proof. ⊓⊔ The following Lemma claims that under (a1)-(a2) exponential stability is really anasymptotic property in the sense that if a l ( k ) or X ( n, k ) is changed on any finite segment n ≤ k ≤ n , this does not influence stability properties of (1). Lemma 4
If for some n > n there exist L , µ , L > , < µ < , such that | X ( n, k ) | ≤ L µ n − k , n ≥ k ≥ n , (9) then for some L , L > , inequality (8) is satisfied. Proof.
Denote A = µ n − n n Y j = n m X l =1 | a l ( j ) | ! . (10)For n ≤ k ≤ n ≤ n we have n − k ≤ n − n and 0 < µ <
1, so Lemma 3 implies | X ( n, k ) | ≤ n − Y j = k m X l =1 | a l ( j ) | ! ≤ Aµ n − n ≤ Aµ n − k , n ≤ k ≤ n ≤ n . If n ≤ k ≤ n , n > n , then by (4), (9), (a1) and (a2) we have h l ( j ) ≥ n , j > n + T ,thus (the second sum below involves only such X ( h l ( j ) , k ) that h l ( j ) < n ) | X ( n, k ) | ≤ | X ( n, n ) || X ( n , k ) | + n + T X j = n | X ( n, j + 1) | m X l =1 | a l ( j ) || X ( h l ( j ) , k ) |≤ L µ n − n Aµ n − k + n + T X j = n L µ n − ( j +1) m X l =1 | a l ( j ) | A ! ≤ AL µ n − k + KmAL n + T X j = n µ n − ( j +1) = AL µ n − k + KmAL µ n − n T X j =0 µ − ( j +1) ≤ Bµ n − k , where B = AL + KmAL µ n − n T X j =0 µ − ( j +1) does not depend on n , A was defined in (10).Choosing L = max { A, B } , we obtain estimate (8) for n ≥ k ≥ n . ⊓⊔ Consider the equation x ( n + 1) − x ( n ) = − m X l =1 b l ( n ) x ( h l ( n )) , n ≥ n . (11)5 orollary 1 Suppose a l ( n ) = b l ( n ) , n ≥ n > n . Equation (11) is exponentially stable ifand only if (1) is exponentially stable. Proof.
Let X ( n, k ) and Y ( n, k ) be fundamental functions of (1), and (11), respectively.Suppose (1) is exponentially stable. Then for X ( t, s ) inequality (8) holds with L > , <µ <
1. Since X ( n, k ) = Y ( n, k ) for n ≥ k ≥ n , then | Y ( n, k ) | ≤ Lµ n − k , n ≥ k ≥ n . By Lemma 4 we have | Y ( n, k ) | ≤ L µ n − k , n ≥ k ≥ n , for some L >
0. Finally, Lemma2 implies that equation(11) is exponentially stable.The same argument proves the converse statement. ⊓⊔ Now we proceed to a result on exponential stability of equation (1) with a positivefundamental function and nonnegative coefficients.
Theorem 1
Suppose the fundamental function of (1) is eventually positive, i.e., for some n ≥ we have X ( n, k ) > , n ≥ k ≥ n , and, in addition, either a = lim inf n →∞ m X l =1 a l ( n ) > or a more general condition holds:there exists a positive integer p , such that b = lim sup n →∞ n + p − Y j = n − m X l =1 a l ( j ) ! < . (13) Then each of the following statements is valid.1. The fundamental function of (1) is eventually nonincreasing.2. If (12) holds, then (8) is satisfied for any µ , − a < µ < ; if (13) holds, thenexponential estimate (8) is valid for any µ, b /p < µ < .3. Equation (1) is exponentially stable. Proof.
By the assumptions of the theorem for n ≥ n + T , k ≥ n we have X ( n + 1 , k ) = X ( n, k ) − m X l =1 a l ( n ) X ( h l ( n ) , k ) ≤ X ( n, k ) , since a l ( n ), X ( h l ( n ) , k ) are nonnegative, so X ( n, k ) is nonincreasing for n ≥ k ≥ n + T ,which completes the proof of the first part of the theorem.Further, let us assume that (12) holds, then there exist ε , 0 < ε < a , and n ≥ n + T such that m X l =1 a l ( n ) ≥ ε, n ≥ n . X ( n, k ) is nonincreasing in n for any fixed k , n ≥ k ≥ n . Consequently, for n ≥ k + T we have X ( h l ( n ) , k ) ≥ X ( n, k ) and X ( n + 1 , k ) ≤ X ( n, k ) − m X l =1 a l ( n ) X ( n, k ) = − m X l =1 a l ( n ) ! X ( n, k ) . The latter inequality yields that 0 ≤ m X l =1 a l ( n ) < n ≥ n = n + T , which implies a ≤ X ( k + 1 , k ) < (1 − ε ) X ( k, k ) = 1 − ε . Repeating this procedure, we obtain X ( n, k ) ≤ µ n − k , n ≥ k ≥ n , where µ = 1 − ε . By Lemma 4 estimate (8) is valid for n ≥ k ≥ n , with the same µ andsome L > q < n ≥ n , where n ≥ n + T . Then the corresponding estimate for any positive integer is X ( k + rp, k ) ≤ q r , so X ( n, k ) ≤ Lµ n − k , n ≥ k ≥ n , where µ = q /p , L = µ − p . Lemma 4 implies the exponential estimate for the fundamental function, n ≥ k ≥ n , thusthe proof of the second part is complete.Finally, by Lemma 2 equation (1) is exponentially stable. ⊓⊔ Remark 1.
Let us remark that a ≤ n = n + T . If a = 1, then an exponential estimate with any positive µ willwork, which is illustrated by the following example. Example 1
For the equation x ( n + 1) − x ( n ) = − (cid:18) − n + 1 (cid:19) x ( n ) , n ≥ , we have a = 1 , where a is defined in (12), the fundamental function is X ( n, k ) = k ! /n ! . Forthis equation exponential estimate (8) is valid for any µ , < µ < , since k ! n ! ≤ µ n − k , n ≥ k ≥ µ . Remark 2.
Let us notice that under the assumptions of Theorem 1 inequality (12) implies(13) for any positive integer p and for b in (13) which does not exceed (1 − a ) p < p such that c = lim inf n →∞ max n ≤ k ≤ n + p − m X l =1 a l ( k ) > . (14)7f (14) holds then for any ε > n large enough among p successive sets of coefficientsat least one satisfies m X l =1 a l ( n ) > cp − ε , so (13) is valid with b ≤ − c/p < Lemma 5 [ ] Suppose a l ( n ) ≥ , l = 1 , , · · · , m , and for some n ≥ n ≥ n m X l =1 a l ( n ) < , sup n ≥ n m X l =1 n − X k =max { n , min l h l ( n ) } a l ( k ) ≤ . (15) Then the fundamental function of (1) is eventually positive: X ( n, k ) > , n ≥ n . We remark ([11], Theorem 7.2.1) that the condition a ≤ k k ( k + 1) ( k +1) (16)is necessary and sufficient for nonoscillation of the autonomous equation x ( n + 1) − x ( n ) = − ax ( n − k ) , k ≥ . (17) Corollary 2
Suppose a l ( n ) ≥ , (13) holds and (15) is satisfied for some n ≥ . Then (1)is exponentially stable. Proof.
By Lemma 5 inequalities (15) imply that the fundamental function of (1) is eventu-ally positive. Application of Theorem 1 completes the proof. ⊓⊔ Consider together with (1) the following comparison equation x ( n + 1) − x ( n ) = − m X l =1 b l ( n ) x ( g l ( n )) , n ≥ n , (18)where g l ( n ) ≤ n . Denote by Y ( n, k ) the fundamental function of equation (18). Lemma 6 [ ] Suppose a l ( n ) ≥ b l ( n ) ≥ , g l ( n ) ≥ h l ( n ) , l = 1 , , · · · , m , for sufficientlylarge n . If equation (1) has an eventually positive solution, then (18) has an eventuallypositive solution and its fundamental function Y ( n, k ) is eventually positive. Corollary 3
Suppose (13) and at least one of the following conditions hold:1) ≤ a l ( n ) ≤ α l , h l ( n ) ≥ n − τ l and there exists λ > such that λ − ≤ − m X l =1 α l λ − τ l . (19) m = 1 , n − h ( n ) ≤ k, a ( n ) ≤ k k ( k +1) ( k +1) .Then (1) is exponentially stable. roof. Suppose λ > f ( λ ) = λ − m X l =1 α l λ − τ l .We have f ( λ ) ≤ , f (1) ≥
0, hence there exists a positive solution λ > x ( n + 1) − x ( n ) = − m X l =1 α l x ( n − τ l ) , which consequently has a positive solution x ( n ) = λ n . Lemma 6 implies that (1) also has apositive solution. By Theorem 1 equation (1) is exponentially stable.Proof of the second part is similar. ⊓⊔ Remark 3.
By Theorem 3.1 in [3] it is enough to assume the existence of an eventuallypositive solution in the conditions of Theorem 1 rather than to require that the fundamentalfunction is positive.
Lemma 1 claims that if (1) is in some sense close to an exponentially stable equation, then itis also exponentially stable. Lemma 7 provides some estimates which are useful to establishthis closeness.Further, we deduce explicit exponential stability conditions based on Lemmas 1 and 7.As above, we assume that (a1)-(a2) hold for (1).
Lemma 7
Suppose the fundamental function of the equation (1) is positive: X ( n, k ) > , n ≥ k ≥ n , and a l ( n ) ≥ , l = 1 , · · · , m , n ≥ n . Then there exists n ≥ n such that ≤ n − X k = n X ( n, k + 1) m X l =1 a l ( k ) ≤ , n ≥ n . (20) Proof.
Since a l ( n ) ≥ X ( n, k ) > n ≥ k ≥ n , then X ( n, k ) is nonincreasing in n for any k , n ≥ k ≥ n . Consider z ( n ) = ( , n ≥ n , , n < n . (21)Then z ( n + 1) − z ( n ) + m X l =1 a l ( n ) z ( h l ( n )) = f ( n ) , n ≥ n , with f ( n ) = m X l =1 a l ( n ) χ n ( h k ( n )), where χ n ( j ) = ( , j ≥ n, , j < n. .Thus by the solution representation (4) for n > n we have1 = z ( n ) = X ( n, n ) + n − X k = n X ( n, k + 1) f ( k ) = X ( n, n ) + n − X k = n X ( n, k + 1) m X l =1 a l ( k ) χ n ( h l ( k )) . T such that χ n ( h l ( k )) = 1 for any l = 1 , · · · , m and k ≥ n + T . Thus for k ≥ n + T we have0 ≤ n − X k = n X ( n, k + 1) n X l =1 a l ( k ) = 1 − X ( n, n ) < . Since X ( n, k ) is nonincreasing and positive, then 0 < X ( n, n ) ≤ n > n . Thus0 ≤ n − X k = n X ( n, k + 1) m X l =1 a l ( k ) ≤ , n ≥ n = n + T , which completes the proof. ⊓⊔ Now let us proceed to explicit stability conditions.
Theorem 2
Suppose there exists a subset of indices I ⊂ { , , · · · , m } such that a k ≥ , k ∈ I , for the sum X l ∈ I a l ( n ) inequality (13) holds, the fundamental function X ( n, k ) of theequation x ( n + 1) − x ( n ) + X l ∈ I a l ( n ) x ( h l ( n )) = 0 (22) is eventually positive and lim sup n →∞ P l I | a l ( n ) | P l ∈ I a l ( n ) < . (23) Then equation (1) is exponentially stable.
Proof.
We apply the same method as in [2], where the comparison (model) equation is (22). ⊓⊔ Remark 4.
Most known explicit stability results include estimates where coefficients a l ( k )are summed up in k from h l ( n ) to n . We note that if the comparison equation is x ( n + 1) − x ( n ) = − b ( n ) x ( n − k ) (24)then the upper index is n − k − Corollary 4
Suppose there exist a set of indices I ⊂ { , , · · · , m } , functions g l ( n ) ≤ n, l ∈ I , and positive numbers α , α and γ < , such that for n sufficiently large the inequalities < α ≤ X l ∈ I a l ( n ) ≤ α < , l ∈ I, hold and the difference equation x ( n + 1) − x ( n ) = − X l ∈ I a l ( n ) x ( g l ( n )) (25)10 as a positive fundamental function. If X k ∈ I | a k ( n ) | max { h k ( n ) ,g k ( n ) }− X j =min { h k ( n ) ,g k ( n ) } m X l =1 | a l ( j ) | + X k I | a k ( n ) | ≤ γ X k ∈ I a k ( n ) (26) then (1) is exponentially stable. Proof.
Let us rewrite (1) in the form x ( n + 1) − x ( n ) = − X k ∈ I a k ( n ) x ( g k ( n )) + X k ∈ I a k ( n )[ x ( g k ( n )) − x ( h k ( n ))] − X k I a k ( n ) x ( h k ( n ))= − X k ∈ I a k ( n ) x ( g k ( n )) + X k ∈ I a k ( n ) σ k max { h k ( n ) ,g k ( n ) }− X j =min { h k ( n ) ,g k ( n ) } [ x ( j + 1) − x ( j )] − X k I a k ( n ) x ( h k ( n ))= − X k ∈ I a k ( n ) x ( g k ( n )) − X k ∈ I a k ( n ) σ k max { h k ( n ) ,g k ( n ) }− X j =min { h k ( n ) ,g k ( n ) } m X l =1 a l ( j ) x ( h l ( j )) − X k I a k ( n ) x ( h k ( n )) , where σ k = ( , g k ( n ) > h k ( n ) , − g k ( n ) < h k ( n ) . Since (25) has a positive fundamental function, thenthe reference to Theorem 2 completes the proof. ⊓⊔ Remark 5.
Based on the choice of subset I , the theorem contains 2 m − I = { , , · · · , m } in Corollary 4, we obtain the following result. Corollary 5
Suppose there exists g ( n ) ≤ n and positive numbers α , α , γ < such thatfor n sufficiently large < α ≤ b ( n ) := m X l =1 a l ( n ) ≤ α < and the difference equation x ( n + 1) − x ( n ) = − m X l =1 a l ( n ) x ( g ( n )) (28) has a positive fundamental function. If for n large enough m X l =1 | a l ( n ) | max { h l ( n ) ,g ( n ) }− X k =min { h l ( n ) ,g ( n ) } m X l =1 | a l ( k ) | ≤ γ m X l =1 a l ( n ) , then equation (1) is exponentially stable. The following result is an immediate corollary of Lemma 5 and Theorem 2.11 orollary 6
Suppose < a ≤ a ( n ) ≤ b < and there exists γ such that < γ < and m X l =1 | a l ( n ) | ≤ γa ( n ) for n large enough. Then the equation x ( n + 1) − x ( n ) = − a ( n ) x ( n − − m X l =1 a l ( n ) x ( h l ( n )) (29) is exponentially stable. Corollary 7
Suppose for some positive a , b , γ , where b < , γ < , the following inequal-ities are satisfied for n large enough < a ≤ m X l =1 a l ( n ) ≤ b < / , (30) m X k =1 | a k ( n ) | n − X j = h k ( n ) m X l =1 | a k ( j ) | ≤ γ m X l =1 a l ( n ) . Then equation (1) is exponentially stable.
Now let us consider the case m = 2 x ( n + 1) − x ( n ) = − a ( n ) x ( g ( n )) − b ( n ) x ( h ( n )) . (31) Corollary 8
Suppose there exist a > and γ , < γ < such that at least one of thefollowing conditions holds for n sufficiently large:1) < a < a ( n ) ≤ a < , n − X k = g ( n ) a ( k ) ≤ , | b ( n ) | ≤ γa ( n ); < a ≤ a ( n ) + b ( n ) ≤ a < , n − X k = g ( n ) ( a ( k ) + b ( k )) ≤ , | a ( n ) | max { h ( n ) ,g ( n ) }− X k =min { h ( n ) ,g ( n ) } [ | a ( k ) | + | b ( k ) | ] < γ [ a ( n ) + b ( n )] . Then equation (31) is exponentially stable.
Proof.
We choose the following equations: x ( n + 1) − x ( n ) = − a ( n ) x ( g ( n )), x ( n + 1) − x ( n ) = − a ( n ) x ( g ( n )) − b ( n ) x ( g ( n )),with a positive fundamental function to obtain 1) and 2), respectively. ⊓⊔ Consider now an autonomous equation with two delays x ( n + 1) − x ( n ) = − ax ( n − g ) − bx ( n − h ) , ag = 0 , bh = 0 . (32)We further apply the nonoscillation condition (16).12 orollary 9 Suppose at least one of the following conditions holds:1) < a ≤ g g ( g + 1) − ( g +1) , | b | < a ;2) < ( a + b ) ≤ g g ( g + 1) − ( g +1) , | a ( g − h ) | < ;Then equation (32) is exponentially stable. Consider a high order autonomous difference equation. x ( n + 1) − x ( n ) = − m X l =1 a l x ( n − l ) . (33) Corollary 10
Suppose there exists k ≥ such that < k X l =1 a l ≤ k k ( k + 1) ( k +1) , m X l = k +1 | a l | < k X l =1 a l . Then equation (33) is exponentially stable.
The present work continues our previous publication [2] where a first order exponentiallystable model x ( n + 1) − x ( n ) = − b ( n ) x ( n ), 0 < ε ≤ b ( n ) ≤ γ < close to the considered equation. Unlike the present paper, [20] considersnonlinear perturbations of stable linear equations as well. The main result (Theorem 2) of[20] is the following one. Suppose that the fundamental function of (1) satisfies n − X j =0 | X ( n, j + 1) | ≤ L, n = n , n + 1 , · · · . (34) Then the nonlinear equation x ( n + 1) − x ( n ) = − m X k =1 a k ( n ) x ( h k ( n )) + F ( n, x ( n ) , x ( n − , · · · , x ( n − l )) is globally asymptotically stable if in addition | F ( n, x , x , · · · , x l ) | ≤ q max ≤ i ≤ l | x i | , and q < L − . Instead of inequality (34) in this paper we apply exponential estimation (8). Gener-ally, (8) implies (34), however for bounded delays and coefficients inequalities (8) and (34)are equivalent. Indeed, solution representation (4) and inequality (34) imply that for any13ounded right hand side | f ( n ) | ≤ M the solution of the problem (2),(3) with the zero initialconditions ( ϕ ( n ) ≡ x ( n ) = 0) is bounded: | x ( n ) | ≤ LM . Thus by the Bohl-Perrontheorem (see Theorem 2 in [2]) the fundamental function satisfies (8), see also Lemma 3 in[20] which claims exponential decay of solutions for autonomous equation (22) if (34) holdsand coefficients a k are positive.Let us discuss some stability tests for equation (1).We start with the following result [9, 19, 23, 24]. If m = 1 , P ∞ n =0 a ( n ) = ∞ , n − h ( n ) ≤ k , a ( n ) ≥ , and n X i = h ( n ) a ( i ) <
32 + 12 k + 2 , (35) then equation (1) is asymptotically stable. This result is also true for general equation (1)( m > ), where a l ( n ) ≥ , a ( n ) = P ml =1 a l ( n ) , h ( n ) = max h l ( n ) . Equation (1) with positive constant coefficients is asymptotically stable if [10] m X l =1 a l lim sup n →∞ ( n − h l ( n )) < e − m X l =1 a l . (36)Stability tests (35) and (36) are obtained for equations with positive coefficients. In thepresent paper we consider coefficients of arbitrary signs. The next interesting feature of theresults obtained here is that some of the delays can be arbitrarily large (see for example,Parts 1 and 2 of Corollary 8). Example 2
By Corollary 8, Part 1, the following two equations x ( n + 1) − x ( n ) = − (0 . .
05 sin n ) x ( n − − . | cos n | x ( n − , (37) x ( n + 1) − x ( n ) = − [0 .
12 + 0 . − n ] x ( n − − [0 . . − n ] x ( n −
14) (38) are exponentially stable.Two previous results of [9, 19, 23, 24] and [10] fail to establish exponential stability forequation (37) with positive coefficients, as well as all parts of Corollary 3.10 in [4] cannot beapplied to equation (38) with an oscillating coefficient. None of the inequalities in Corollary8 of [2] can be applied to establish stability of (38).We note that (37),(38) are special cases of equation with one nondelay term and twodelay terms considered in [2], however none of the inequalities in Corollary 8 of [2] can beapplied to establish stability of (38). Let us also note that for (38) we have n − X k = g ( n ) | a ( k ) | + n − X j = h ( n ) | b ( j ) | = n − X k = n − [0 .
12 + 0 . − k ] + n − X j = n − (cid:12)(cid:12)(cid:12) . . − k (cid:12)(cid:12)(cid:12) = 1 . > π , where [13] π/ is the best possible constant [19, 22] in P mk =1 ka k < π/ which implies expo-nential stability of (33). xample 3 Consider equation (31) with two variable coefficients and delays, where a ( n ) = ( − . , if n is even , − . , if n is odd , b ( n ) = ( . , if n is even , . , if n is odd , (39) g ( n ) = ( n − , if n is even ,n − , if n is odd h ( n ) = ( n − , if n is even ,n − , if n is odd. (40) Then in (31) the sum a ( n ) + b ( n ) is either 0.05 or 0.03 which is less than 0.5, P n − k = n − [ a ( k ) + b ( k )] ≤ · .
05 + 2 · .
03 = 0 . < / for both odd and even n and for γ = 0 . < we have | a ( n ) | max { h ( n ) ,g ( n ) }− X k =min { h ( n ) ,g ( n ) } [ | a ( k ) | + | b ( k ) | ] = ( . · .
29 = 0 . < . γ, n is even , . . · .
29] = 0 . < . γ, n is odd , thus by Corollary 8, Part 3, equation (31) is exponentially stable. None of the inequalitiesin Corollary 8 of [2] can be applied to establish stability of (31). We note that it would beharder to treat this example if the equation were written as high order equations with constantdelays and variable coefficients. A number of papers [2, 12, 14, 15, 16, 17, 20] are devoted to stability tests for equationswith positive and negative coefficients and, more generally, for equations with oscillatingcoefficients. Paper [20] extends earlier results of [12]. In particular, for the linear autonomousequation x ( n + 1) − x ( n ) = qx ( n − m ) − px ( n − k ) , p > , q > , m ≥ , k ≥ . (41)the following result was obtained in [12]. Suppose p ( k +1) ( k +1) k k ≤ . Then equation(41) is exponentially stable if and only if p > q . Condition 1) of Corollary 9 is close to this result. It gives the same sufficient stabilitytest for q of an arbitrary sign but does not involve the necessity part.The paper [16] contains a nice review on stability results obtained for equations withoscillating coefficients. The results of [16] generalized the following stability test obtained in[14] for equation (41): If kp < , p − kp kp > q then (41) is asymptotically stable. By condition 2) of Corollary 9 equation (41) is asymptotically stable if p − q < k k ( k + 1) ( k +1) , | p ( k − m ) | < . It is easy to see that these two tests are independent.Let us discuss sharpness of conditions of Theorem 1 for exponential stability of (1),assuming the fundamental function is positive; in particular, we demonstrate sharpness ofcondition (12). 15 xample 4
The equation x ( n + 1) − x ( n ) = − − n − x ( n ) , n ≥ n ≥ , has a positive fundamental function and any solution can be presented as x ( n ) = x ( n ) n − Y k = n (1 − − k − ) , thus for n = 0 X ( n,
0) = n − Y k =0 (1 − − k − ) > − n − X k =0 − k − > − ∞ X k =0 − k − = 12 , i.e., the equation is neither asymptotically nor exponentially stable. Let us demonstrate that the facts that the sum of coefficients m X l =1 a l ( n ) in (1) is posi-tive, exceeds a positive number and that the fundamental function is positive do not implystability, in the case when coefficients have different signs. Example 5
Consider the difference equation x ( n + 1) − x ( n ) = − . x ( n −
1) + 2 x ( n ) . (42) Here . − . > , so the sum of coefficients exceeds a certain positive number. Let usprove that the fundamental function is positive and the solution is unbounded. Really, for thefundamental function we have X (0 ,
0) = 1 , X (1 ,
0) = 3 . Denote x ( n ) = X ( n, , notice that x (1) > . x (0) and prove x ( n ) > . x ( n − > by induction. Really, x ( n ) > . x ( n − > implies x ( n − < x ( n ) / , and for any x ( n − > we have x ( n + 1) = 3 x ( n ) − . x ( n − > x ( n ) − . x ( n ) / . x ( n ) > . x ( n ) , thus X ( n, is positive and unbounded. The equation is autonomous, so the same is true for X ( n, k ) . Since X ( n, is unbounded, then (42) is not stable. Finally, let us formulate some open problems.1. Under which conditions will exponential stability of (1) imply exponential stability ofthe equation with the same coefficients and smaller delays: x ( n + 1) − x ( n ) = − m X l =1 a l ( n ) x ( g l ( n )) , n ≥ n , h l ( n ) ≤ g l ( n ) ≤ n ?16. Prove or disprove:If in Theorem 2 condition (23) is substituted by X l I | a l ( n ) | ≤ α n X l ∈ I a l ( n ) , ∞ Y n =1 α n ≤ ∞ X n =1 (1 − α n ) = ∞ , then (1) is asymptotically stable.3. Consider the problem of the exponential stability of (1) when (a1)-(a2) are substitutedwith one of two more general conditions: a) lim n →∞ h l ( n ) = ∞ and there exists M > n X j = h l ( n ) a l ( j ) < M, for any l = 1 , · · · , m ; b) delays are infinite but coefficients decay exponentially with memory, i.e., there existpositive numbers M and λ < | a l ( n ) | ≤ M λ n − h l ( n ) .Let us note that Bohl-Perron type result in case b) was obtained in [5], Theorem 4.7.4. Example 5 demonstrates that for equations with positive and negative coefficients anda positive fundamental function inequality (12) does not imply exponential stability.Is it possible to find such conditions on delays and coefficients of different signs that(12) would imply exponential stability? For instance, prove or disprove the followingconjecture. Conjecture.
Suppose the following conditions a ( n ) ≥ b ( n ) ≥ , h ( n ) ≤ g ( n ) ≤ n, lim sup n →∞ b ( n )[ g ( n ) − h ( n )] < x ( n + 1) − x ( n ) = − a ( n ) x ( h ( n )) + b ( n ) x ( g ( n )) . (43)If the fundamental function of (43) is positive andlim inf n →∞ [ a ( n ) − b ( n )] > , then (43) is exponentially stable.Let us remark that conditions when the fundamental function of (43) is positive wereobtained in [6]. Acknowledgement
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